Mathematical model of positive column of electric discharge in the nondiffusion approximation

The theory of positive column of gas discharge in the plasma approximation is developed with due regard for the ion inertia, ionization friction, and directed component of the ion velocity in the frequency of collisions between ions and atoms. In the diffusion approximation, these factors are not taken into account. The problem is reduced to two essentially nonlinear first-order differential equations for the velocity and concentration of ions. In dimensionless variables, the behavior of distributions of the velocity and concentration, as well as the characteristics of the positive column, depends on two parameters, namely, the number of ion collisions over the characteristic length of the system and the ratio of this length to the Debye length. It is shown that the transition to the Schottky theory takes place at the values of parameters exceeding $10^4$. In discharges of low and moderate pressure, the use of ambipolar diffusion approximation may, in some cases, lead to a considerable error. The characteristics of the positive column in a wide range of variation of parameters are calculated.